State Space Model of a Sructure

7/29/2019 State Space Model of a Sructure

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State Space Model of a Structure

R. Kashani, Ph.D.www.deicon.com

Traditionally, structural dynamics has been formulated using a set of secondorder ODEs. As fa-

miliar to the practitioners as this formulation is, it has some drawbacks. For example, decoupling the

differential equations of a multiDOF system with viscous damping using this formulation is not possible.

Moreover, this formulation does not lend itself to active multiinputmultioutput control either. These

limitations are addressed using state space formulation, in which a set of first order ODEs describe the

system dynamics, depicted in the generic form of

x Ax

Bu

y Cx

Du

where x, u, and y are the vectors of states, inputs, and outputs, respectively. A, B, C, and D are the

constant dynamics, input, output, and direct transmission matrices, respectively. In structural dynamics,

displacements and velocities (in physical and modal domains) are the most commonly used states. Readers

unfamiliar with the state space formulation of dynamics systems are referred to State Space Modeling

tutorial.

The formulation presented by Equations 1 and 2 is the basis for statespace modeling of flexible

structures, having point force(s) as the input(s) and point displacement(s) as the measured output(s).

z

O I 2 2

z

O

Q

u (1)

y W O z

Du (2)

where

state vector: z t = t

t

number of modes: Nm

number of inputs: Nunumber of outputs: Ny

modal displacement: t = 1 t 2 t Nm t T

modal velocity: t = 1 t 2 t Nm t T

input: u t = u1 t u2 t uNu t T

spatial coordinate i: ri

output: y t = x r1 t x r2 t x rNy t T

natural frequency: = diag 1 2 Nm

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modal damping: = diag 1

2 ! ! !

Nm "

eigenfunction i at location j : i j

modal input matrix: Q = #$%

1 & 1 ' ' ' 1 & Nu...

. . ....

Nm

& 1 ' ' ' Nm

& Nu

( )

0

modal output matrix: W = #$%

1 & 1 ' ' ' Nm & 1...

. . ....

1 & Ny ' ' ' Nm & Ny

( )

0

The A, B, C, and D matrices describing the flexible structures statespace model, shown in Equation

1 and 2, are functions of the parameters of the system (natural frequencies, damping ratios, and mode

shapes, i.e., 1 2 i i and i 4 i 5 1 6 6 6 n,) resulting in the following model format

z 1 A 7 8 z 9 B 7 8 u (3)

y 1 C7 8 z 9 D 7 8 u (4)

Input/Output Formulation of the Equation of Motion

Without the loss of generality, the structural state space formulation is carried out for a one dimensional

structure, i.e., a beam.

Input Matrix

When the beam is excited by a point force p0 and a point moment m0, the modal equation of motion is

i 7 t8 9 2iii 7 t8 9 2i i 7 t8 1 Q

mi 7 r8 m0 7 t8 9 Q

pi 7 r8 p0 7 t8 (5)

where Qmi 7 r8 and Qpi 7 r8 are the generalized modal forcing functions due to point moment and point force

inputs, respectively. The general expansion of this forcing function is

Qi 7 r8 1 7 i 7 r8 F7 r8 8 1 CL

0i 7 r8 F7 r8 dr (6)

where L is the length of the beam and F7 r8 could be viewed as the spatial representation of the forcing

function; see Equations (7) and (8).

When input to the beam is a point force p07

t8

acting at location r1

a, the forcing function f7

r t8

is

f 7 r

t8 1 p0 7 t8 7 r E a 8F G H I

FP rQ

(7)

and when the input is a point moment m0 7 t8 acting at location r 1 b, the forcing function is

f 7 r

t81

m0 7 t8 R 7 r E b 8F G H I

FP rQ

(8)

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The generalized modal forcing functions, using Equations (7) and (8) are

Qpi S rT U W

L

0i

S

rT S

r Y a T dr

U iS

a T (9)

QmiS

rTU Y W

L

0i

S

rT aS

r Y b T dr

UY i

S

rT S

r Y b T bb

b

L

0 cW

L

0ai

S

rT S

r Y b T dr

U

c

aiS

b T (10)

Thus

iS

tTc

2iiiS

tTc

2i iS

tTU Y

aiS

b T m0S

tTc

iS

a T p0S

tT (11)

Equation (11) confined to a finite number of modes is the basis for state space modeling of a beam/structure.

When a pair of point moments of equal magnitude and opposite sign, resembling piezoelectricpatch

actuation, act at r1 and r2, Equation (11) becomes

iS

tTc

2i iS

tT U Y e aiS

r2 T Y aiS

r1 T h m0S

tT (12)

Output Matrix

Displacement/velocity measurement When the measured output is displacement or velocity, Equation

13 or its derivative is used to formulate the output equation.

wS

Pi tTU

n q 1

WnS

P T nS

tT (13)

Note that D in the output equation is a zero matrix.

Example 0.2: Model of a 2-DOF system

Formulate the state space model of the 2DOF springmassdashpot system shown in Figure 1. The

x1

Fr

c1

k1

m1s

c2

k2

m2s

x2

Figure 1: A 2DOF discrete system

force F exciting the mass m1 is the input and the displacement of mass m2, i.e., x2, is the ouput of the

system. The following parameters are considered in the model: k1 U 5, k2 U 10, m1 U 8, and m2 U 15.

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The massnormalized eigenvalues are 1t

u

0 w 1673 0 w 2274 xy and 2

tu

0 w 3114 0 w 1222 xy .

The natural frequencies are 1t

0 w 4197 and 2t

1 w 5380. In the absence of damping the natural frequen-

cies are used to construct the A matrix as

At

O I

2

2

t

0 0 1 0

0 0 0 1

0w

4197

2

0 0 00 1 w 5380 2 0 0

(14)

The input and output matrices are formulated using the elements of the eigenvectors corresponding to

mass m1 where the excitation is inputing the system and mass m2 where the displacement is measured.

Bt

O

Q

t

0

0

0 w 1673

0 w 3114

(15)

and Ct

W O t

0 w 2274 0 w 1222 0 0 (16)

Strain measurement: Strain, measured by a patch of piezoelectric ceramic, at any small region of a

structure is proportional to the curvature at that region. In a beam the curvature is the second derivative of

the displacement, i.e.,

rt

2y

r2(17)

t

2

r2

i 1

i r i t (18)

where , the proportionality factor, is the distance from the neutral axis of the beam. For an individualmode i

i r t i t2i r

r2(19)

where is a constant. The signal generated by a patch of piezoelectric strain gauge installed along thelength of the beam between r

t

r1 and r2 is proportional to the integral of strain over the domain of the

gauge, i.e. between r1 and r2. So the contribution of the mode i to measured output is

r2

r1 i

r

t

i

t

r2

r1

2i r

r2 dr (20)

t

i ti r

r

r2

r1

t

i r2 i r1 i t

t

Wii t (21)

Thus the element of the W matrix mapping the modal coordinate system to the measured signal by the

strain measuring device is proportional to i r2 i r1 . Note that this is the same as the element of

the Q matrix transforming the physical piezoelectricpatch actuation, represented by a pair of moments of

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equal magnitude and opposite signs, to the mode i. For collocated arrangements W QT, which similarto the relationship between Q and W when point force is the input and displacement is the output, with

collocated arrangement.

The model structure described above is based on the classical (proportional) damping assumption, i.e.,

damping coefficient matrix is proportional to the mass M and stiffness K matrices. Although this is more

of a mathematical convenience than physical reality, the assumption is widely used in modeling flexible

structures. In the absence of classical damping assumption, the statespace model of the flexible structurecan still be represented in terms of eigenvalues and eigenvectors of the system.

Model Structure of Systems with NonProportional Viscous Damping

The equation of motion of an ndegreeoffreedom system in physical coordinates is:

Mj x k

Cj x k

Kj x F (22)

Multiplying Equation (22) by Ml 1 and rewriting it in state space yields

z m

O Io Ml 1K o Ml 1C

z km

O

Ml 1 u (23)

where X t x t

x t is the state vector. This system in the absence of rigid body motion and repeated

eigenvalues has n pairs of complex conjugate eigenvalues and their corresponding complex conjugate

eigenvectors:

1 1 1 1 2 2 2 2 { z z z

Transforming the coordinates of the system fromX to using the transformation matrix P, i.e.,y

P,defined asP | I1

R1

I2

R2 z z z } (24)

results in

P l 1AP k P l 1Bu (25)

where the superscripts I and R signify imaginary and real parts of the complex i. P l1AP is the

block diagonal real matrix, shown below, with block entries consisting of real and imaginary parts of the

eigenvalues

P l 1AP blockdiag m

R1o I1

I1 R1

m

R2o I2

I2 R2

z z z

(26)

The output equation is

y P (27)

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State Space Model of a Sructure